U.S. patent number 6,632,014 [Application Number 09/729,526] was granted by the patent office on 2003-10-14 for device and method for mixing substances.
This patent grant is currently assigned to Yeda Research and Development Co., Ltd.. Invention is credited to Alexander Groisman, Victor Steinberg.
United States Patent |
6,632,014 |
Steinberg , et al. |
October 14, 2003 |
Device and method for mixing substances
Abstract
A method and device for mixing a liquid with another substance
are presented. The mixing is based on creation of a turbulent flow
of the liquid, by providing curvilinear trajectories of the flow
and providing a polymer material in the liquid flow.
Inventors: |
Steinberg; Victor (Rehovot,
IL), Groisman; Alexander (Rehovot, IL) |
Assignee: |
Yeda Research and Development Co.,
Ltd. (Rehovot, IL)
|
Family
ID: |
24931453 |
Appl.
No.: |
09/729,526 |
Filed: |
December 4, 2000 |
Current U.S.
Class: |
366/315; 366/338;
366/348 |
Current CPC
Class: |
B01F
3/10 (20130101); B01F 5/0646 (20130101); B01F
5/0647 (20130101); B01F 7/26 (20130101); B01F
2005/0025 (20130101) |
Current International
Class: |
B01F
3/10 (20060101); B01F 3/08 (20060101); B01F
7/26 (20060101); B01F 5/06 (20060101); B01F
5/00 (20060101); B01F 005/06 (); B01F 007/26 () |
Field of
Search: |
;366/338,341,315,316,339,336,348,349 |
References Cited
[Referenced By]
U.S. Patent Documents
Other References
Shraiman, B.I. & Siggia, E.D. Scalar turbulence, Nature, 405,
639-645 (Jun. 2000). .
Warhaft, Z, Passive scalars in turbulent flows Annu.Rev. Fluid
Mech. 32,203-240 (2002). .
Bird, R.B., Curtiss, C.F., Armstrong, R.C. & Hassager, O.,
Dynamics of polymeric liquids, John Wiley, NY, 1987. .
R.G. Larson et al., "A Purely Viscoelastic Instability in
Taylor-Coutte Flow", J Fluid Mech., 218, 573-600, (Jul. 6, 1989).
.
Byars, J.A., Oztekin, A., Brown R.A. & McKinley, G.H., Spiral
instabilities in the flow of highly elastic fluids between rotating
parallel disks, J. Fluid Mech., 271, 173-218 (Jan. 24, 1994). .
Joo, J.L. & Shaqfeh, E.S.G., Observations of purely elastic
instabilities in the Taylor-Dean flow of a Boger fluid, J. Fluid
Mech. 262, 27-73 (Aug. 23, 1993). .
A. Groisman and V.Steinberg, "Elastic Turbulence in Polymer
Solution Flow", Nature 405, pp. 53-55, 2000..
|
Primary Examiner: Soohoo; Tony G.
Attorney, Agent or Firm: Browdy and Neimark PLLC
Claims
What is claimed is:
1. A method of creating an elastic turbulent flow of a liquid, the
method comprising: providing a flow of said liquid with curvilinear
trajectories of the flow, and providing in said flow of liquid an
elastic polymer material soluble in said liquid wherein the flow is
defined by the elasticity of the polymer.
2. The method according to claim 1, wherein the concentration of
the polymer material is at least 0.001%.
3. The method according to claim 1, wherein said polymer material
is a flexible high molecular weight polymer of 10.sup.6
g.multidot.mol.sup.31 1 or higher.
4. The method according to claim 1, and further comprising the step
of supplying a substance into said flow with the curvilinear
trajectories, thereby enabling mixing of said liquid with the
substance.
5. The method according to claim 4, wherein the mixing is carried
at low Reynolds numbers up to 0.06 or less.
6. A method of mixing substances, at least one of the substances
being a liquid, the method comprising): (i) providing a continuous
flow of the substances with curvilinear trajectories of the flow;
and (ii) providing in the liquid flow an elastic polymer material
soluble in the liquid, thereby creating elastic turbulence of the
flow defined by the elasticity of the polymer and wherein the
elastic turbulence of the liquid is substantially irrespective of a
Reynolds number of the flow.
7. The method according to claim 6, wherein the concentration of
the polymer material is at least 0.001%.
8. The method according to claim 6, wherein said polymer material
is a flexible high molecular weight polymer of 10.sup.6
g.multidot.mol.sup.31 1 or higher.
9. The method according to claim 6, wherein the mixing is carried
out at low Reynolds numbers up to 0.06 or less.
10. The method according to claim 6, wherein said flow is a
continuous flow of a solution of the substances with the polymer
along an open-end curvilinear channel.
11. The method according to claim 10, wherein said channel defines
a serpentine-like path.
12. The method according to claim 10, wherein said channel defines
a worm-like path.
13. The method according to claim 6, wherein said flow is a
closed-loop continuous flow of the solution of the substances with
the polymer with the curvilinear trajectories of the flow.
14. A mixing device for mixing substances, at least one of the
substances being a liquid, the mixing device comprising: a mixing
tank of cylindrical shape having upper and lower disks, a space
between the disks forming a mixing channel for a flow of the
substances with a polymer material soluble in the liquid, at least
one of the upper and lower disks being mounted for rotation to
thereby provide a closed loop flow of the solution of the
substances with the polymer in the channel with curvilinear
trajectories of the flow.
15. A mixing device for mixing substances, at least one of the
substances being a liquid, the mixing device comprising: (a) a
mixing channel for a flow of the substances therein, the channel
being formed in a mixing tank of a cylindrical shape having upper
and lower disks for the substances to be supplied into a space
between the disks, at least one of the upper and lower disks being
mounted for rotation, the rotation of the at least one of the disks
providing a closed-loop continuous flow of the substances in the
channel with curvilinear trajectories of the flow; and (b) a supply
means for supplying the substances into the channel with presence
of a polymer material soluble in the liquid.
16. A method of mixing substances, at least one of the substances
being a liquid, the method comprising: providing a solution of the
substances with a polymer soluble in said liquid in a
cylindrically-shaped space between two disks; and rotating at least
one of the disks, thereby providing a continuous flow of said
solution of the substances with the polymer in said space with
curvilinear trajectories of the flow, said flow being turbulent
thereby assisting in the mixing of the substances.
17. A method of creating a turbulent flow of a liquid, the method
comprising: providing a flow of said liquid with curvilinear
trajectories of the flow, and providing in said flow of liquid a
polymer material, which is soluble in said liquid and is a flexible
high molecular weight polymer of 10.sup.6 g.multidot.mol.sup.31 1
or higher.
Description
FIELD OF THE INVENTION
This invention relates to a device and method for mixing
substances, particularly very viscous substances in small
volumes.
BACKGROUND OF THE INVENTION
The mixing of liquids is essential for many industrial and
laboratory processes, and has been addressed in the past, for
example in the following publications: (1) Shraiman, B. I. &
Siggia, E. D. Scalar turbulence, Nature, 405, 639-545 (2000). (2)
Warhaft, Z., Passive scalars in turbulent flows Annu. Rev. Fluid
Mech. 32, 203-240 (2000).
Since the process of molecular diffusion is typically characterized
by a long characteristic time, rapid mixing almost always requires
some macroscopic flow, which is regularly induced by stirring or
shaking. In order to provide efficient mixing, however, the flow
needs to be chaotic or turbulent. It is known that a flow is likely
to be turbulent, when the Reynolds number, Re, is large
(Re=VL/.nu., wherein V is the liquid velocity, L is the size of a
tank in which the liquid flows, and .nu. is the kinematic viscosity
of the liquid). Thus, in order to obtain a high Reynolds number,
the liquid velocity and the tank size should be sufficiently large
while the liquid should be of low viscosity. When the liquids are
very viscous and/or the tank is small, the velocity required to
create a turbulent flow may be so high, that it becomes quite
impractical. In this case, liquids arc usually mixed in closed
mixers. However, this interrupts the continuous technological
processes and requires a lot of energy to provide a homogeneous
mixture.
It is known that solutions of flexible high molecular weight
polymers differ from newtonian fluids in many aspects. The most
notable elastic property of the polymer solution is that stress
does not immediately become zero, when the fluid motion stops, but
rather decays with some characteristic time, .lambda., which can
reach seconds and even minutes. The equation of motion for dilute
polymer solutions differs from the Navier-Strokes equation defining
the motion of simple, low molecular weight newtonian fluids by an
additional linear term arising from the elastic stress. Since the
elastic stress is caused by stretching of the polymer coils, it
depends on history of motion and deformations of fluid elements
along their flow trajectories. This implies a nonlinear
relationship between the elastic stress and the rate of strain in
the flow. These features can be learned from the following
publication: (3) Bird, R. B., Curtiss, C. F., Armstrong, R. C.
& Hassager, O., Dynamics of polymeric liquids, John Wiley, NY,
1987.
The non-linear mechanical properties of viscoelastic fluids can
lead to many special flow effects, such as purely elastic
transitions that quantitatively change character of the flow at
vanishingly small Reynolds number. This is disclosed in the
following publications: (4) R. G. Larson et al., "A Purely
Viscoelastic Instability in Taylor-Couette Flow", J. Fluid Mech.,
218, 573-600, 1990; (5) Byars, J. A., Oztekin, A., Brown R. A.
& McKinley, G. H., Spiral instabilities in the flow of highly
elastic fluids between rotating parallel disks, J. Fluid Mech.,
271, 173-218 (1994). (6) Joo, J. L. & Shaqfeh., E. S. G.,
Observations of purely elastic instabilities in the Taylor-Dean
flow of a Boger fluid, J. Fluid Mech 262, 27-73 (1994).
As a result of such transitions, secondary vortical flow appears in
different systems, where the primary motion is a curvilinear shear
flow. The onset of those secondary flows depends on the Weissenberg
number, Wi, determined as Wi=.lambda..gamma., wherein .lambda. is
the polymer relaxation time, and .gamma. is the shear rate. The
Weissenberg number plays a role analogous to that of the Reynolds
number in competition between non-linearity and dissipation.
SUMMARY OF THE INVENTION
There is a need in the art to facilitate the mixing of substances,
by providing a novel method and device that enables the efficient
mixing of substances even very viscous, in small volumes, at
arbitrary low Reynolds numbers. The present invention provides for
the gentle mixing of viscous liquids in small size channels at low
velocities and small applied stresses, as well as mixing between a
viscous liquid and a powder.
It has been found by the inventors that the flow of a sufficiently
elastic polymer solution can become very irregular even at low
velocity, high viscosity, and in a small volume (tank). The fluid
motion is excited in a broad range of spatial and temporal scales,
and the flow resistance significantly increases (by a factor up to
twenty), thereby presenting a turbulent flow. These main features
of turbulence appear in a flow of a highly elastic polymer
solution, even at arbitrarily low Reynolds numbers. A comparable
state of turbulent flow for a newtonian fluid in a pipe would have
a Reynolds number as high as 10.sup.5.
The inventors have found that the nonlinearity of mechanical
properties of a fluid can give rise to a turbulent flow when the
equation of motion is linear. For a polymer solution, this
corresponds to a state in which the Weissenberg number is high,
while the Reynolds number is small. This situation can be realized
if the parameter of elasticity, Wi/Re=.lambda..nu./L.sup.2, is
large enough, wherein L is characteristic size and .nu. is
kinematic viscosity of the fluid.
The main idea of the present invention is based on the creation of
turbulence in a liquid (even very viscous liquid) in a flow with
curvilinear trajectories, by adding a small amount of polymer. This
can be used for mixing this liquid with another substance (liquid
or powder). The flow of an elastic polymer solution at sufficiently
high values of Weissenberg number, Wi, has all the main features of
the developed turbulence. The increase in the flow resistance
resulting in the turbulence of the flow is due to the elastic
stress provided by the presence of a polymer material.
There is thus provided according to one aspect of the present
invention a method of creating a turbulent flow of a liquid, the
method comprising the step of providing a polymer material in the
liquid flow with curvilinear trajectories.
For the purposes of the present invention, the presence of a
polymer material of at least 0.001% concentration is sufficient.
Preferably, the polymer material is a flexible high molecular
weight polymer.
The above technique can be used for effective mixing of the liquid
with another substance (liquid or powder). The efficient mixing can
be carried out at arbitrary small Reynolds numbers.
According to another aspect of the present invention, there is
provided a method of mixing substances, at least one of the
substances being a liquid, the method comprising the steps of: (i)
providing a continuous flow of the substances with curvilinear
trajectories of the flow: and (ii) providing a polymer material in
the liquid flow, thereby creating turbulence of the flow.
To provide effective mixing of the substances, the flow
periodically turns, resulting in that the difference in the
concentration of the substances in the flow exponentially reduces.
A characteristic length of the path defining effective mixing of
the substances is preferably such that this difference reduces by
about 3 times.
According to some embodiments of the invention, the curvilinear
trajectories of the flow are achieved by directing the flow along a
serpentine- or worm-like channel, so as to provide an open
continuous flow of the substances through the channel between inlet
and outlet openings thereof. According to another embodiment of the
invention, the curvilinear trajectories of the flow are achieved by
circulating the substances in a cylindrically shaped mixing tank.
Such a tank defines a closed continuous flow of the substances with
the curvilinear trajectories of the flow.
According to yet another aspect of the present invention, there is
provided a mixing device for mixing substances, at least one of the
substances being a liquid, the mixing device comprising: (a) a
mixing tank for a flow of the substances therein with curvilinear
trajectories of the flow; and (b) a supply means for supplying the
substances into the tank with presence of a polymer material in the
liquid flow.
BRIEF DESCRIPTION OF THE DRAWINGS
In order to understand the invention and to see how it may be
carried out in practice, a preferred embodiment will now be
described, by way of non-limiting example only, with reference to
the accompanying drawings, in which:
FIGS. 1a to 1c schematically illustrate three examples of a mixing
tank suitable to be used in a mixing device according to the
invention;
FIG. 2 graphically illustrates a stress ratio as a function of a
shear rate as obtained with the mixing tank of FIG. 1a;
FIG. 3 illustrates two snapshots of the flow obtained in the tanks
of FIG. 1a, showing turbulence of the flow at Wi=13 and Re=0.7;
FIGS. 4a and 4b illustrate snapshots presenting the experimental
results obtained with the mixing tank of FIG. 1b;
FIG. 5 illustrates the distribution of the power of velocity
fluctuations in the middle of the channel of FIG. 1b;
FIGS. 6a to 6d illustrate space-time plots of mixing a polymer
solution at different positions along the channel;
FIG. 7 illustrates plots of PDF of concentration of a fluorescent
dye at different positions in the channel;
FIG. 8 illustrates the dependence of the moments of distribution on
the position along the channel; and
FIG. 9 illustrates correlation coefficients for the concentration
as functions of the distance across the channel.
DETAILED DESCRIPTION OF THE INVENTION
Referring to FIG. 1a, there is illustrated an example of a device 1
suitable to be used as a mixing device according to the invention.
The device 1 comprises a cylindrical cup 2 (constituting a mixing
tank) having upper and lower concentric plates 2a and 2b (i.e.,
parallel disks) for a liquid to be supplied into a spaced
therebetween until a level d. The upper plate 2a is mounted for
rotation being attached to a rotating shaft 4, and just touches the
surface of the liquid to provide a swirling flow of the liquid
between the disks. A special cover is used to minimize evaporation
of the liquid. The space between the disks 2a and 2b thus presents
a cavity for a closed continuous flow of a liquid thereinside with
curvilinear trajectories of the flow.
Following is an example illustrating the creation of a turbulent
flow of a liquid in the tank 2. In the present example, the radii
of the upper and the lower plates are R.sub.1 =38 mm and R.sub.2
=43.6 mm, respectively. The liquid used is a solution of 65%
saccharose and 1% NaCl in water, viscosity .eta..sub.s =0.324
Pa.multidot.s, as a solvent for added polymer, which is
polyacrylamide (M.sub.W =18,000,000; Polysciences) at a
concentration of 80 p.p.m by weight. The viscosity of the
so-obtained solution is .eta.=0.424 Pa.multidot.s at .gamma.=1
s.sup.-1. The curvature ratio is quite high, d/R=0.263, to provide
destabilization of the primary shear flow and development of the
secondary vortical fluid motion at lower shear rates.
For experimental purposes, the whole flow set-up 1 is mounted on
top of a commercial viscometer (AR-1000 of TA-instruments) to
measure precisely the angular velocity .omega., of the rotating
upper plate and the torque applied to it, to estimate the average
shear stress in a polymer solution flowing inside the cup. The
walls of the cup 2 are transparent, which allows Doppler
velocimeter measurements by collecting light scattered from the
crossing point of two horizontal laser beams. The flow is observed
from below. The lower plate 2b of the cup is made from plexiglass,
and a mirror (not shown) tilted by 45.degree. is placed under the
lower plate. The flow patterns are then captured by a CCD camera
(photodetector) at the side, and the temperature is stabilized at
12.degree. C. by circulating air in a closed box. The flow under
the black upper plate is visualized by seeding the liquid with
light reflecting flakes (1% of the Kalliroscope liquid). The liquid
is illuminated by ambient light. The relaxation time, .lambda.,
estimated from the phase shift between the stress and the shear
rate in oscillatory tests, was 3.4 s.
FIG. 2 graphically illustrates the measurement results, showing
stress ratio as functions of shear rate. The ratio of the average
stress .sigma., measured in the flow to the stress .sigma..sub.lam
in a laminar flow with the same boundary conditions is plotted as a
function of the shear rate, .gamma.. Two curves C.sub.1 and C.sub.2
correspond to the polymer solution flow with d.sub.j =10 mm and
d.sub.2 =20 mm, respectively. The shear rate was gradually varied
in time, very slowly (by about 10% h.sup.-1) in the transition
region, and faster below and above it. In the graphs, thin black
lines represent increasing .gamma.; and thick gray lines represent
decreasing .gamma.. Curve C.sub.3 represents the pure solvent. As
shown, mechanical degradation of the polymers is quite small at
shear rates below 1.5 s.sup.-1 and 1 s.sup.-1 for d=10 mm and 20
mm, respectively. The dependence of .sigma./.sigma..sub.lam on
.gamma. in those regions were therefore reproducible in consecutive
runs within about 1%. Degradation effects became appreciable at
higher shear rates, and elasticity typically decreased by up to 10%
as a result of the runs shown by curves C.sub.1 and C.sub.2.
FIG. 3 shows two snapshots of the flow at Wi=13, Re=0.7. As shown,
although the pattern is quite irregular, structures that appear
tend to have spiral-like forms. The dark spot in the middle
corresponds to the center of a big persistent thoroidal vortex that
has dimensions of the whole set-up.
Thus, the experimental results have shown that by adding a high
molecular weight polymer into a liquid, and providing curvilinear
trajectories of a flow of the liquid with polymer, the turbulence
of flow can be obtained. More particulars of the above experiment
can be learned from the following article: A. Groisman and V.
Steinberg, "Elastic Turbulence in a Polymer Solution Flow", Nature,
405, 53-55, 2000. The disclosure in this article is therefore
incorporated herein by reference.
The cup 2 can thus be used as a mixing tank for mixing two
substances, wherein one of the substances is a liquid containing a
polymer (e.g., the above indicated solution). The mixing tank 2 is
of a kind providing a closed continuous motion of the substances,
such that the polymer-containing liquid moves along a circular
trajectory inside the tank. It has been found that effective mixing
is achieved on a length of the liquid path of about 100 times of
the distance between the disks 2a and 2b. The degree of mixing is
almost independent of the size of the tank, viscosity of the liquid
and the flow velocity, which can be very low.
Referring now to FIGS. 1b and 1c, there are illustrated two more
examples of a mixing tank suitable to be used in the present
invention for mixing substances. In each of these examples, a
mixing device utilizes a mixing tank of a kind providing an open
continuous flow of substances thereinside. In a device 10 (FIG. 1b)
a mixing tank 12 is configures to define a curvilinear channel of a
serpentine shape, while a mixing tank 112 (FIG. 1c) defines a
worm-like channel.
More specifically, the tank 12 of FIG. 1b is composed of a sequence
of N smoothly connected half-rings (units), generally at R, having
outer and inner diameters R.sub.1 and R.sub.2 of the ring. The path
providing effecting mixing is defined by the characteristic length
of the channel, which is about 50-100 times of the channel's width
d. For experimental purposes, the channel is formed with an inlet
opening 12A for feeding therein working substances L.sub.1 and
L.sub.2 to be mixed (both being liquids in the present example),
and an outlet opening 12B for discharging a resulting mixture S
therefrom. The liquids L.sub.1 and L.sub.2 are fed into the channel
by two syringe pumps (which are not specifically shown) at equal
rates through two separate tubes 14A and 14B, respectively. The
pumps and tubes constitute together a supply arrangement. To
provide a turbulent flow of at least one of the liquids, a small
amount of a polymer material (0.001% is sufficient) is added to
this liquid.
Following is an example of a mixing technique carried out in the
channel 12. In the present example, the following conditions are
used. The liquids are identical, each containing a solution of 65%
saccharose and 1% NaCl in water, with the viscosity .eta..sub.s
=0.153 Pa.multidot.s and density .rho.=1.32 g/cm.sup.3, as a
solvent for the polymer. The polymer, which in the present example
is added to both liquids, is polyacrylamide (M.sub.W =18,000,000;
Polysciences). One of the solutions is also added with c.sub.0 =2
p.p.m. of a fluorescent dye (fluorescene), used for measurement
purposes, as will be clear from the description below. The solution
viscosity is .eta.=0.198 Pa.multidot.s at a shear rate .gamma.=4
s.sup.-1.
The channel of a depth d=3 mm is machined in a transparent bar of
perspex and scaled from above by a transparent window. The outer
and inner diameters r.sub.1 and r.sub.2 of the half-rings are,
respectively, of 3 mm and 6 mm. The channel is square in the
cross-section and has 30 repeating units, each with a linear
dimension of 18 mm.
The experiment is carried out at a room temperature, i.e.,
22.5.+-.0.5.degree. C. The total rate of the liquid supply, Q, into
the channel was always kept constant, so that the average time of
mixing inside the channel was proportional to the position N along
the channel.
For the measurement purposes, the channel is illuminated from a
side by an Argon-Ion laser beam converted by two cylindrical lenses
to a broad sheet of light with a thickness of about 40 .mu.m in the
region of observation. The fluorescent light emitted by the liquid
in the perpendicular direction is projected onto a CCD camera and
digitized by a 8-bit 512.times.512 frame grabber. Concentration of
the dye is evaluated from the intensity of the light, which was
found to be proportional to the concentration.
The flow is always observed near the middle of the half-ring close
to the side from which the laser beam comes. Hence, the number N of
the unit is a natural linear coordinate along the channel.
The relaxation time, .lambda., estimated from the phase shift
between the stress and the shear rate in oscillatory tests is 1.4
s. An estimate for the diffusion coefficient of the dye is given by
that for the saccharose molecules, which is about
D=8.5.multidot.10.sup.-7 cm.sup.2 /s. The characteristic shear
rate, .gamma., and the Weissenberg number W.sub.i in the flow are
estimated as follows: ##EQU1##
The Reynolds number, Re=2Q.rho./(d.eta.) was always quite low,
reaching 0.6 for the highest value of Q in the experiment.
Referring to FIGS. 4a and 4b, there are shown snapshots of the flow
at N=29 as imaged by the CCD camera during the liquids flow along
the region of observation. FIG. 4a shows the situation for a pure
solvent at Re=0.16, and FIG. 4b shows the situation for the polymer
solution at the same flow rate, corresponding to Wi=6.7. In the
figures, bright regions correspond to high concentrations of the
fluorescent dye.
As shown in FIG. 4a, the small Reynolds number results in that the
flow of the pure solvent remained quite laminar and no mixing
occurred. The boundary that separates the liquid with and without
the dye is smooth and parallel to the direction of the flow, and it
became smeared due to molecular diffusion as the liquid advances
downstream. Behavior of the polymer solution was qualitatively
different from that of the solvent. The flow was laminar and
stationary only up to a value of Q corresponding to Wi.sub.c =3.2
(and Re=0.06), at which an elastic instability occurred. As shown
in FIG. 4b, this instability leads to irregular flow and mixing of
the liquids. The experiments were carried out at Q about twice
above the flow instability onset, Wi=6.7, at which homogeneity of
the mixture at the exit of the channel was the highest.
Turning now to FIG. 5, there are illustrated three graphs G.sub.1,
G.sub.2 and G.sub.3 showing power spectra of fluctuations of the
flow velocity in the middle of the channel at N=12 and Wi=6.7,
i.e., P(f). Graph G.sub.1 corresponds to the spectra of the
velocity components for the polymer solution along the mean flow,
graph G.sub.2 corresponds to the same across the mean flow, and
graph G.sub.3 --for the pure solvent across the mean flow. The flow
velocity was measured by a laser Doppler anemometer, when the
region of the laser beam crossing was made very small, i.e.,
15.times.15.times.40 .mu.m, in order to decrease the gradient
noise. The mean velocity for the polymer solution was V=6.6 mm/s.
The RMS of the fluctuations, V.sub.rms, was 0.09V and 0.04V for,
respectively, longitudinal and transversal directions. The spectra
of both longitudinal and transversal velocity components do not
exhibit any distinct peaks and have broad regions of a power decay,
which is typical for turbulent flow.
Mixing of the polymer solution is a random process, and may
therefore be characterized statistically by a probability
distribution function (PDF) in order to find different
concentrations, c, of the dye in a point, and by values of the
moments, M.sub.i, of the distribution. The i.sup.th moment is
defined as an average <.vertline.c-c.sub.1.vertline..sup.i
>/c.sub.1.sup.i, wherein c.sub.1 is the average concentration of
the dye, which in the present example is equal to c.sub.0 /2,
wherein c.sub.0 is the initial concentration of the dye. Small
values of the moments M.sub.i signify high homogeneity and good
mixing of the liquids.
Reference is made to FIGS. 6a-6b, 7 and 8. FIGS. 6a and 6b
illustrate space-time diagrams of the flow taken at different
positions along the channel corresponding to different values of
M.sub.1, i.e. M.sub.1 =0.72, N.sub.1 =8 and M.sub.1 =0.25, N.sub.2
=24, respectively. The brightness profile was captured 12.5 times
per second along a single line across the channel near the middle
of a half-ring. Profiles measured at consecutive moments of time
are plotted as horizontal lines from top to bottom.
FIG. 7 illustrates PDF of the concentration of the fluorescent dye
at different positions, wherein each graph represents statistics
over about 10.sup.7 points corresponding to about 50 space-time
diagrams, and a total liquid discharge of 2.multidot.10.sup.3
d.sup.3. The regions near the walls of the channel with the width
of 0.1 d were excluded from the statistics. Graphs H.sub.1 and
H.sub.2 in the figure correspond to the situations of FIGS. 6a and
6b.
FIG. 8 illustrates dependences of the M.sub.1 and M.sub.2 values
(represented by dark symbols) on the position N along the channel.
The average flow time t.sub.0 is connected to N as follows: t.sub.0
=N.multidot.7.8s.
In order to observe further stages of mixing, a series of
experiments were carried out, where the liquids were premixed
before they entered the channel. For these purposes, a shorter
channel with the same shape was used and accommodated upstream of
the channel 12, such that the liquids were first passed through the
shorter channel and then entered the channel 12. The experimental
results are shown in FIGS. 6c-6d, FIG. 7 (graphs H.sub.3 and
H.sub.4) and FIG. 8 (light symbols). The space-time plots (FIGS. 6c
and 6d) were taken at positions corresponding to M.sub.1 =0.082
(N.sub.3 =39), and M.sub.1 =0.030 (N.sub.4 =54), respectively.
As a result of premixing, PDF of the dye concentrations at N=2 was
almost identical to PDF at N=27 without the premixing. Hence, in
FIG. 8, the values M.sub.1 and M.sub.2 for the flow with the
premixing are plotted on the same graph adding a number of 25 to
the position N along the channel. It is seen that the curves
plotted for the liquids premixing case are indeed continuations of
the dependences obtained for the values M.sub.1 and M.sub.2 in the
channel without the premixing.
Thus, as the liquid flows downstream, it becomes increasingly
homogeneous and PDF of the dye concentration becomes narrower. As
shown, in FIGS. 6a and 7 (graph H.sub.1), there are large
homogeneous regions with maximal and zero dye concentration, and
PDF has a maximal value near c.sub.0 and zero. The dependences on
the entrance condition fades gradually as the liquids flow
downstream and get mixed. Therefore, the space-time diagram in FIG.
6b has a lot of fine scale structures of different brightnesses.
The corresponding PDF (FIG. 7, graph H.sub.2) has a single peak at
c.sub.1, and long tails that decay exponentially and touch the
limits of the concentration, zero and c.sub.0. Further downstream,
the space time diagram (FIG. 6c corresponding to N=39) exhibits
characteristic features at similar spatial scale, but are much more
faded. The PDF (FIG. 7, graph H.sub.3) is much narrower and has
quite clear exponential tails, which imply strong intermittency in
mixing. The distribution is well confined in a region far from the
limits of zero and c.sub.0. Hence, the dependence on the initial
condition should be quite minor by that point. At the last point
(FIG. 6d), the non-homogeneity in the concentration is hardly seen,
and the PDF (FIG. 7, graph H.sub.4) is very narrow.
FIG. 9 illustrates representative spatial autocorrelation functions
for the dye concentration, namely, the correlation coefficients for
the concentration as functions of the distance .DELTA..sub.x across
the channel. Here, graphs S.sub.1-S.sub.4 correspond to the graphs
H.sub.1 -H.sub.4 of FIG. 7 and to the space-time plots in FIGS.
6a-6d, respectively. At large N (N>29), the correlation
functions at different positions become identical.
It is evident from FIG. 8, that both M.sub.1 and M.sub.2 decay
exponentially above N=30, the rate of the decay being two times
higher for M.sub.2 than that for M.sub.1. The higher order moments
were found to decay exponentially, M.sub.i.about.exp(-.gamma..sub.i
N), as well.
Turning back to FIG. 5, spatial structure of the flow in the
channel can be inferred from the power spectra shown in the figure,
if the Taylor hypothesis disclosed in the above publications (1)
and (2) is applied. This spectra imply that the power of the
velocity fluctuations scales with the k-number in space as
P.about.k.sup.-3.3. Fluctuations of the velocity gradients should
thus scale as k.sup.-1.3, so that the flow becomes increasingly
homogeneous at small scales, and mixing is mainly due to the
largest eddies having the size of the whole flow system.
In an another example of the present invention, a more concentrated
sugar syrup (as compared to that used in the previously described
example) was used as a solvent, and a polymer solution was prepared
with viscosity and relaxation time about two times larger man those
of the original solution. With this polymer solution, substantially
the same efficiency of mixing was obtained at corresponding Wi,
while characteristic flow rates were twice lower, and Re was about
four times lower. Dependence of the efficiency of mixing at the
optimal flow conditions on concentration of the polymers was very
weals (although Wi.sub.c grew fast, when the polymer concentration
was decreasing). Hence, for a solution with the polymer
concentration of 10 p.p.m. (.eta./.eta..sub.s =1.03), M.sub.1 of as
low as 0.22 was reached at N=29 (and at Re=0.065). The mixing was
observed down to the polymer concentration of 7 p.p.m.
The advantages of the present invention are thus self-evident. By
providing turbulence of the flow of a liquid by adding it with a
polymer material, it can be easily and efficiently mixed with
another liquid or powder. Very viscous liquids can be efficiently
mixed at very low flow rates with the aid of polymer additives at
very low concentrations.
Those skilled in the art will readily appreciate that various
modifications and changes can be applied to the preferred
embodiment of the invention as hereinbefore exemplified without
departing from its scope defined in and by the appended claims.
* * * * *